Control System Analysis on Symmetric Cones

Transcription

1 Submitted, 015 Conference on Decision and Control (CDC) Control System Analysis on Symmetric Cones Ivan Papusha Richard M. Murray Abstract Motivated by the desire to analyze high dimensional control systems without explicitly forming computationally expensive linear matrix inequality (LMI) constraints, we seek to exploit special structure in the dynamics matrix. By using Jordan algebraic techniques we show how to analyze continuous time linear dynamical systems whose dynamics are exponentially invariant with respect to a symmetric cone. This allows us to characterize the families of Lyapunov functions that suffice to verify the stability of such systems. We highlight, from a computational viewpoint, a class of systems for which stability verification can be cast as a second order cone program (SOCP), and show how the same framework reduces to linear programming (LP) when the system is internally positive, and to semidefinite programming (SDP) when the system has no special structure. I. INTRODUCTION In this paper we wish to verify the stability of a linear dynamical system ẋ(t) Ax(t), A R n n, x(t) R n. (1) It is well known that if A has no special structure, then system (1) is stable if and only if there exists a symmetric matrix P P T R n n satisfying the linear matrix inequalities P P T 0, A T P PA 0. () Such a matrix P corresponds to a quadratic Lyapunov function V : R n R, V (x) x, P x, which is positive definite and decreasing along trajectories of (1). Searching for a matrix P satisfying () is in general a semidefinite program (SDP), but if A has a special structure, this semidefinite program can sometimes be cast as a linear program (LP) with advantages in numerical stability, opportunities for parallelism, and better scaling to high dimensions than the general SDP. For example, if A is a Metzler matrix, i.e., its off-diagonal entries are nonnegative, A ij 0, for all i j, then it suffices to search for a diagonal matrix P satisfying (). Equivalently, it is enough to find an entrywise positive vector p int R n such that Ap int R n is entrywise positive. Finding such a vector p is, of course, an LP, [1], [], [3]. The authors are with the Control and Dynamical Systems Department, California Institute of Technology, Pasadena, CA USA. Autonomous systems for which A is a Metzler matrix are called internally positive, because their state space trajectories are guaranteed to remain in the nonnegative orthant R n if they start in the nonnegative orthant. As we will see, the Metzler structure is (in a certain sense) the only natural matrix structure for which a linear program may generically be composed to verify stability. However, the inclusion LP SOCP SDP motivates us to determine if stability analysis can be cast, for example, as a second order cone program (SOCP) for some specific subclass of linear dynamics, in the same way that it can be cast as an LP for internally positive systems, or an SDP for general linear systems. Using Jordan algebraic techniques, which have proved to be effective in unifying interior point methods for conic programming, we will discuss a generalization of the Metzler property known as cross-positivity to characterize dynamics that are exponentially invariant with respect to a symmetric cone. In so doing, we demonstrate the strong unifying power of the Jordan product on linear systems and discuss a rich class of systems that admit SOCP-based analysis. II. JORDAN ALGEBRAS AND SYMMETRIC CONES The following background is informal and mostly meant to set out notation, adopting conventions familiar to system theory and optimization [4], [5]. Jordan algebras in the context of symmetric cones are rich and well studied field, with an excellent reference [6]. For more recent reviews, see [7], [8], [9]. A. Cones on a vector space. Let V be a vector space with inner product,. Asubset K V is called a cone if it is closed under nonnegative scalar multiplication: for every x K and θ 0 we have θx K. AconeK is pointed if it contains no line, or equivalently x K, x K x 0. Aconeisproper if it is closed, convex, pointed, and has nonempty interior. Every proper cone induces a partial order on V given by x, y V, x y y x K. We have x y if and only if y x int K. Similarly we write x y and x y to mean y x and y x, respectively.

2 B. Jordan algebras. A(Euclidean)Jordanalgebraisaninnerproductspace (V,, ) endowed with a Jordan product : V V V, which satisfies the following properties: 1) Bilinear: x y is linear in x for fixed y and vice-versa ) Commutative: x y y x 3) Jordan identity: x (y x) (x y) x 4) Adjoint identity: x, y z y x, z. In particular, a Jordan product need not be associative. When we interpret x x x, thejordanidentityallowsanypower x k, k to be inductively defined. An identity element e satisfies e x x e x for all x V,anddefinesa number r e, e called the rank of V. The cone of squares K corresponding to the Jordan product is defined as K {x x x V }. Every element x V has a spectral decomposition x λ i e i, where λ i R are eigenvalues of x and the set of eigenvectors {e 1,...,e r } V,calledaJordan frame, satisfies e i e i, e i e j 0for i j, e i e. We have x K (written x K 0,orsimplyx 0 when the context is clear) provided λ i 0. Similarly,x int K if and only if λ i > 0 (written x K 0). The spectral decomposition allows us to define familiar concepts like trace, determinant, and square root of an element x by taking the corresponding function of the eigenvalue. That is, r tr x λ i, det x λ i, / λ 1/ i e i, where the last quantity only makes sense if x 0. Finally, every element z V has a quadratic representation, which is a map P z : V V given by C. Symmetric cones. P z (x) (z (z x)) z x. The closed dual cone of K is defined as K {y V x, y 0, for all x K}, AconeK is self-dual if K K. Theautomorphismgroup Aut(K) of an open convex cone K is the set of invertible linear transformations g : V V that map K to itself, Aut(K) {g GL(V ) gk K}. An open cone K is homogeneous if Aut(K) acts on the cone transitively, in other words, if for every x, y K there exists g Aut(K) such that gx y. Finally,K is symmetric if it is homogeneous and self-dual. Symmetric cones are an important object of study because they are the cones of squares of a Jordan product, admit a spectral decomposition, and are (in finite dimensions, [6]) isomorphic to a Cartesian product of n n self-adjoint positive semidefinite matrices with real, complex, or quaternion entries, 3 3 self-adjoint positive semidefinite matrices with octonion entries (Albert algebra), and Lorentz cone. In practice, symmetric cones have a differentiable log-det barrier function, allowing numerical optimization via interior point methods [7], [10]. The symmetric cones R n ( S 1 S 1 ), L n and S n give rise to LP, SOCP, and SDP, respectively. We now define these three cones. D. Three important symmetric cones. 1) Nonnegative orthant R n : the cone of squares associated with the standard Euclidean space R n and Jordan product (x y) i x i y i, i 1,...,n, i.e., theentrywise(orhadamard)product.theidentityelement is e 1 (1,...,1), andthejordanframecomprises the standard basis vectors. The quadratic representation of an element z R n is given by the diagonal matrix P z diag(z) x Fig. 1. Second-order (Lorentz) cone L 3. ) Lorentz cone L n : partition every element of R n as x (, ) R R n 1 and define the Lorentz cone (also known as second-order or norm cone) as L n {(, ) R R n 1 } R n. This cone is illustrated in Figure 1 for the case n 3.It can be shown that L n is the cone of squares corresponding to the Jordan product [ x0 ] [ ] [ y0 y 1 x, y y 1 y 0 ]. The rank of this algebra is, giving a particularly simple spectral decomposition x λ 1 e 1 λ e, where the eigenvalues and Jordan frame are λ 1, λ, e 1 1 [ ] 1, e / 1 [ ] 1. /

3 The identity element is e (1, 0) and the quadratic representation of an element z R n is the matrix P z zz T zt J n z J n, where J n diag(1, 1,..., 1) R n n is a n n inertia matrix with signature (1,n 1). NotethatP z is a positive semidefinite matrix if and only if z 0. 3) Positive semidefinite cone S n : consider the vector space S n of real, symmetric n n matrices with the trace inner product. If we define the Jordan product as the symmetrized matrix product, X Y 1 (XY YX), then the cone of squares is S n,thepositivesemidefinite matrices with real entries. The spectral decomposition of an element X S n follows from the eigenvalue decomposition n X λ i v i vi T, e i v i vi T, i 1,...,n. The quadratic representation of an element Z S n is given by the map P Z : S n S n, P Z (X) (Z (Z X)) (Z Z) X ZXZ. III. COMPUTATIONAL APPROACH TO DYNAMICS ON A SYMMETRIC CONE Instead of discussing the original dynamics (1) on R n, we consider instead general linear dynamics on a Jordan algebra V.Thiswillallowustopresentaunifiedtreatmentof the cones R n, L n,aswellasrestatefamiliarresultsfrom classical state space theory by specializing to S n.welist most notable results by Gowda et al. [8], [9], [11]. A. Cross-positive linear operators. A linear operator L : V V is cross-positive with respect to a proper cone K if x K, y K, and x, y 0 L(x),y 0. Exponentials of cross-positive operators leave the cone invariant: if L : V V is cross-positive with respect to K, then e tl (K) K for all t 0. Infact,theconverseisalso true [1, Theorem 3]. For example, a matrix A R n n,thoughtofasalinear transformation A : R n R n,iscross-positivewithrespect to the cone R n if and only if A is a Metzler matrix, i.e., A ij 0, for all i j, or equivalently that trajectories x(t) of the system ẋ(t) Ax(t) remain in R n whenever they enter R n.therefore,crosspositivity can be thought of as a generalization of the Metzler property. Examples of operators L that satisfy e L (K) K are Nonnegative orthant: L(x) Ax, wherea R n n is Metzler. Lorentz cone: L(x) Ax, wherea T J n J n A ξj n for some ξ R. Positive semidefinite matrices: L(X) AX XA T for any matrix A. In fact, due to a result in [13], the examples above precisely characterize cross-positive operators on R n and L n (but not S n ). B. A class of Lyapunov functions Let L : V V be a linear operator, and consider the linear dynamical system ẋ(t) L(x), x(0), (3) where V is an initial condition. We make the assumption that e L (K) K, orequivalently,thatl is crosspositive. Systems that obey this this assumption are, of course, very special, because any trajectory that starts in the cone of squares K will remain within K for all time t 0, K x(t) K for all t 0, in other words, such systems are exponentially invariant with respect to the cone K. Weconsiderageneralizationofthe class of quadratic Lyapunov functions on V given by V z : V R, V z (x) x, P z (x), where z V is a parameter, and P z is the quadratic representation of z in the Jordan algebra V.Thefollowing theorem gives a necessary and sufficient condition that V z is alyapunovfunction Theorem 1 (Gowda et al. 009). Let L : V V be a linear operator on a Jordan algebra V with corresponding symmetric cone of squares K, andassumethate L (K) K. The following statements are equivalent: (a) There exists p K 0 such that L(p) K 0 (b) There exists z K 0 such that LP z P z L T is negative definite on V. (c) The system ẋ(t) L(x) with initial condition K is asymptotically stable. Proof. See [11, Theorem 11]. This result allows the existence of a distinguished element p K 0 in the interior of K, wherethevectorfieldpointsin the direction of K, orequivalently L(p) K 0,tobea certificate for the existence of a quadratic Lyapunov function of the form V z : V R, V z (x) x, P z (x). In fact, the derivative of V z along trajectories of (3) is precisely V z (x) ẋ, P z (x) x, P z (ẋ) L(x),P z (x) x, P z (L(x)) x, L T (P z (x)) x, P z (L(x)) x, (P z L L T P z )(x)

4 For the algebra R n and the corresponding cone R n, Theorem 1 translates as follows. The quadratic representation of an element z R n 0 is a diagonal matrix D with strictly positive diagonal. Let A be a cross-positive (i.e., Metzler) matrix. The system ẋ(t) Ax(t) is stable (i.e., Hurwiz), if and only if there exists an entrywise positive vector p such that p R n 0, Ap R n 0, which happens if and only if there exists a diagonal Lyapunov function V (x) x T Dx, D S n 0, AD DA T S n 0. Finding such a p is a LP for a fixed A. Inaddition,ifwe know (or impose, through a feedback interconnection) that A has the Metzler structure, then a diagonal Lyapunov function candidate suffices to ensure stability. This trick has been used in, e.g., [3] to greatly simplify(and in certain cases parallelize) stability analysis and controller synthesis. Now consider the algebra S n and the corresponding cone S n.onecandefinemanycross-positiveoperators,butone comes to mind: the Lyapunov transformation L : S n S n, L(X) AX XA T, where A R n is a given matrix. (By construction, L is cross-positive). Following the theorem, we now restate some widely known facts about linear systems. The matrix differential equation Ẋ(t) AX XA T, X(0) X 0 S n 0 is asymptotically stable if and only if there exists a matrix P 0 such that L(P )AP PA T 0, ifandonlyif there exists a matrix Z 0 such that the function V Z (X) X, P Z (X) Tr(XZXZ) XZ F is a Lyapunov function. This happens if and only if A is a (Hurwiz) stable matrix. IV. CASE STUDY: DYNAMICS ON A LORENTZ CONE The cones R n and S n have been well studied in the literature. The intermediate case the cone L n is, however, quite strange. This cone has received relatively little attention in the control community. Recent efforts have been in the context of model matching [14]. Our work here can be seen as a complement. A. Enforcing L n -invariance To apply the main theorem, we require that e L (K) K. In the case K L n,thedynamicsmatrixa should apriori satisfy the LMI A T J n J n A ξj n 0, ξ R. (4) If A were to affinely depend on optimization variables (as it would if it were a closed loop matrix in a linear feedback synthesis problem), then enforcing L n -invariance would also be an LMI we might as well dispense with any special structure, and resort to algebraic Riccati, bounded-real, or general LMI-based analysis, e.g., [15]. However, if A has some extra structure, for example, it has an embedded positive block transverse to the Lorentz cone axis, then the LMI (4) can be simplified, as we discuss below after introducing the concept of diagonal dominance. B. Diagonal dominance. A square matrix A R n n is (weakly) diagonally dominant if its entries satisfy A ii j i A ij, for all i 1,...,n. As a simple consequence of Gershgorin s circle theorem, diagonally dominant matrices with nonnegative diagonal entries are positive semidefinite. If there exists a positive diagonal matrix D R n n such that AD is diagonally dominant, then the matrix A is generalized or scaled diagonally dominant. Equivalently, there exists a positive vector y R n 0 such that A ii y i j i A ij y j, for all i 1,...,n. Note that generalized diagonally dominant matrices are also positive semidefinite, and include diagonally dominant matrices as a special case. Symmetric generalized diagonally dominant matrices with nonnegative diagonal entries are also known as H -matrices, and have a very nice characterization as matrices with factor width of at most two, see [16, Theorem 9]. One consequence of this characterization is that H matrices can be written as a sum of positive semidefinite matrices that are nonzero only on a principal submatrix, see [17]. For example, a 3 3 H -matrix A A T is the sum of terms of the form x 0 y 1 0 y A x x z 1 z, z z y 0 y 3 where the sub-matrices are all positive semidefinite, [ ] [ ] [ ] x1 x y1 y 0, z1 z 0, 0. x x 3 y y 3 z z 3 Recently, this fact has been exploited in [18], [19] to extend the reach of sum-of-squares techniques to high dimensional dynamical systems without imposing full LMI constraints on the Gram matrix, and in [17] to preprocess SDPs for numerical stability. C. Rotated quadratic constraints. Real, symmetric, positive semidefinite matrices of size, astheyoccurinthecharacterizationofh -matrices above, are special, because they satisfy a restricted hyperbolic constraint; hence their definiteness can be enforced with

5 asocpratherthansdp[7].specifically,wehave, [ ] x1 x 0 x x 0, x 3 0, x 3 x 0 3 [ x x x 3 3] for scalars, x,andx 3. D. Embedded positive block. ( x 3, x, x 3 ) L 3, We are now equipped to consider the simple, augmented (n 1)-dimensional dynamics ] [ ][ ] [ẋ0 a0 0 x0, a 0 R, A R n n, ẋ 1 with trajectory x(t) ( (t), (t)) R R n,onthe cone L n1.afterwritingoutthelmi(4)inblockform, we determine that the augmented system is L n1 -invariant if and only if a 0 I (A A T ) 0. (5) It is stable if and only if A is Hurwiz and a 0 < 0. Roughly speaking it is L n1 -invariant and stable if the stability degree of A (i.e., minusthemaximumrealpartoftheeigenvalues of A) isatleast a 0. In general, the stability degree constraint (5) is an LMI in the variables (A, a 0 ),however,ifa is Metzler then the LMI can be replaced with the constraint a 0 I (A A T ) is a H -matrix, (6) without any loss. From previous sections, the H -matrix constraint (6) is a SOCP. x where the base dynamics x A x are given by the internally positive system ẋ 1 1 l 31 l ẋ ẋ 3 0 l 1 l 3 l 3 0 x l 31 l 3 l 3 l 43 l 34 x 3. ẋ 4 }{{} } 0 0 {{ l 43 4 l 34 x 4 }}{{} x(t) A x(t) Here, the state x i represents the amount of material at node i, l ij 0 the transfer rate between nodes i and j in the base system, a 0 R the self-degradation rate of the catchall buffer, and h i the rate of transfer between node i and the catch-all node. Note that A is Metzler by construction. Several problems are now readily solved: by checking (5) (or (6)), with the l ij treated as variables and h 0fixed, the augmented system is L 5 -invariant only if a 0 1.5,which is an upper bound on the Metzler eigenvalue of A. Thusthe catch-all node must consume material no faster than with rate constant 1.5. Inaddition,ifl ij and h i are known, and the augmented system (7) is L 5 -invariant, then it is stable provided a 0 < 0. Ofcourse,thesetypesofproblemscanbe cast as LP or SOCP. F. Other examples on L n. x Fig. 3. x Embedded focus along -axis x 4 1) Embedded block: For the dynamics ] [ ][ ] [ẋ0 a0 0 x0, ẋ 1 x 3 Fig.. Transportation network,...,x 4 from [3, Figure ], augmented with a catch-all buffer. E. Example in 5D. We consider the linear transportation network shown in Figure, which is inspired by the one studied in [3]. The network might represent a base system of four buffers (solid nodes,...,x 4 )exchangingandconsumingmaterialina way that preserves the network structure. The base system is augmented with an extra catch-all buffer (grayed out node ), leading to the augmented dynamics [ẋ0 x ] [ a0 h T ][ x0 x ], (t) R, x(t) R 4, (7) This system is L n -invariant if and only if A A T a 0 I. It is stable if and only if A is Hurwiz and a 0 < 0. Figure3 illustrates embedded block dynamics without a shear term h. ) Twist system: Suppose AA T 0(skew-symmetric). The system ] [ ][ ] [ẋ0 a0 h T x0, ẋ 1 is L n -invariant if and only if h a 0, i.e., thepoint (a 0,h) L n.furthermore,ifa 0 0,thisisatwistin homogeneous coordinates. 3) Proper orthochronous Lorentz transformations: The restricted Lorentz group SO (1,n 1), whichisgenerated by spatial rotations and Lorentz boosts, consists of linear transformations that keep the quadratic form x T J n x invariant. In particular, elements of SO (1,n 1) correspond to L n -invariant dynamics matrices.

6 Algebra: Real Lorentz Symmetric V R n R n S n K x, y R n x T y L n x T y S n Tr(XY T ) x y x i y i (x T 1 y, y 1 y 0 ) (XY YX) P z, z int K diag(z) zz T zt J nz ( J n ) X ZXZ V (x) x, P z(x) x T diag(z) x x T zz T zt J nz J n x XZ F Free variables in V (x) n n n(n 1)/ dynamics L x Ax x Ax X AX XA T L is cross-positive A is Metzler A satisfies (4) by construction L(p) K 0 (Ap) i < 0 (Ap) 1 ( Ap) P PA T 0 Stability verification LP SOCP SDP TABLE I SUMMARY OF DYNAMICS PRESERVING A CONE V. CONCLUSION In this work we used Jordan algebraic techniques to analyze linear systems with special structure. In particular, we put analysis of internally positive systems, which have recently been in vogue, in the same theoretical framework as systems that are exponentially invariant with respect to the Lorentz cone, as well as general linear systems. It is evident that Lyapunov functions for such systems can be significantly simpler, computationally and representationally, than if the special cone-invariant dynamic structure were not exploited. In this framework, summarized in Table I, the relevant structure is cross-positivity of the dynamics matrix on a symmetric cone. Unfortunately, while cross-positivity has a fairly simple LP characterization in terms of the corresponding dynamics matrix (via the Metzler structure) for internally positive systems, the same condition is generically much more complicated for L n -invariant dynamics: the condition is a LMI in the dynamics matrix. In fact, the condition corresponding to entrywise positivity of the dynamics matrix is also a LMI for discrete time systems [0], [1]. As a result, L n - invariance, by itself, is not computationally attractive for high dimensional systems without taking advantage of even more special structure. We gave a simple (almost trivial) example of how this might be done by augmenting an internally positive system with a catch-all block. Future work will aim to apply these techniques to control synthesis and as well as to further study special structure. An interesting question is whether it is possible to verify stability of certain subclasses of, e.g., differenceofpositive or ellipsoidal cone invariant systems via LP or SOCP, rather than via SDP. ACKNOWLEDGMENTS We thank S. You, N. Matni, and A. Swaminathan for helpful discussions. This work was supported in part by adepartmentofdefensendsegfellowship,terraswarm, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA, and by the Boeing company. REFERENCES [1] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, ser. Classics in Applied Mathematics. SIAM, [] P. D. Leenheer and D. 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